State aggregation methods
Using the book's random walk example, If you have a state space with $1000$ states and you divide them into $10$ groups, each of those groups will have $100$ neighboring states. The function for approximation will be
\begin{equation}
v(\mathbf w) = x_1w_1 + x_2w_2 + ... + x_{10}w_{10}
\end{equation}
Now, when you pick a state, the feature vector will be a one-hot encoded vector with $1$ that is placed in a position that depends on in which group does the chosen state belong. For example if you have state $990$ that state belongs in group $10$ so the feature vector will be
\begin{equation}
\mathbf x_t = [0, 0, ..., 0, 1]^T
\end{equation}
what this means is that the only weight that will be updated is weight $w_{10}$ because gradients for all other weights will be $0$ (that's because features for those weights are $0$)